**Calculating the Pressure Head at the Lower End of the Pipe**

To calculate the pressure head at the lower end of the pipe, we need to consider the pressure at the upper end, the head loss due to friction, and the average velocity of water flow.

**Pressure at the Upper End**

Given that the pressure at the upper end of the pipe is 5m of water, we can convert this to pressure head by using the equation:

Pressure head = pressure / (density of water x gravitational acceleration)

The density of water is approximately 1000 kg/m³, and the gravitational acceleration is 9.81 m/s². Substituting these values, we have:

Pressure head = 5m / (1000 kg/m³ x 9.81 m/s²) = 0.051 m³/s²

**Head Loss due to Friction**

The head loss due to friction can be calculated using the Darcy-Weisbach equation:

Head loss = (friction factor x length x velocity²) / (diameter x 2g)

Where:
- friction factor: a dimensionless factor that depends on the pipe roughness and Reynolds number
- length: length of the pipe (10 m in this case)
- velocity: average velocity of water flow (5 m/s in this case)
- diameter: diameter of the pipe (1 m in this case)
- g: gravitational acceleration (9.81 m/s²)

To calculate the friction factor, we need to know the pipe roughness and Reynolds number. Let's assume a typical value for the roughness of a commercial steel pipe, which is around 0.045 mm. The Reynolds number can be calculated using the equation:

Reynolds number = (velocity x diameter) / kinematic viscosity

Where:
- velocity: average velocity of water flow (5 m/s in this case)
- diameter: diameter of the pipe (1 m in this case)
- kinematic viscosity: a property of water, which is approximately 1 x 10⁻⁶ m²/s

Substituting the values, we have:

Reynolds number = (5 m/s x 1 m) / (1 x 10⁻⁶ m²/s) = 5 x 10⁶

Using the Moody chart or Colebrook equation, we can find the friction factor corresponding to this Reynolds number and pipe roughness. Let's assume a value of 0.019, which is typical for commercial steel pipes.

Now we can calculate the head loss:

Head loss = (0.019 x 10 m x (5 m/s)²) / (1 m x 2 x 9.81 m/s²) = 0.49 m

**Pressure Head at the Lower End**

The pressure head at the lower end of the pipe can be calculated by subtracting the head loss due to friction from the pressure head at the upper end:

Pressure head at lower end = Pressure head at upper end - Head loss due to friction

Pressure head at lower end = 0.051 m³/s² - 0.49 m = -0.439 m

Since the pressure head is negative, it indicates that the pressure at the lower end of the pipe is lower than atmospheric pressure. This is due to the head loss caused by friction along the length of the pipe.

Therefore, the pressure head at the lower end of the